Hello, we have the same problem. DS is now 14 weeks and he screams if not on me...no moses basket, crib or pram. In the day he naps in the sling, at night on me in bed. I don't sleep well and never intended to co-sleep as I thought it was dangerous but it is the only way DS will sleep.
I asked a MW who does sleep problems about it, and she said it is extremely common and not really a problem per se, more them expressing their need to be close. She said my lack of sleep was more to do with my concerns about safety and to address that either by making baby their own little area on the bed or getting a co-sleeper.
I started a thread about it and learnt on here that some babies do brilliantly in co-sleepers, others somehow know it is not the same bed and will not tolerate them at all even though they are still next to their mums. So I didn't want to spend lots of money just to own a fourth thing that the baby will not sleep in...
One of the suggestions was to buy an Ikea Gulliver cot (£60) because the fourth side is removable and you can improvise a co-sleeper from it, and if DS won't go down in it I can still get use out of it as a proper cot when he is older by re-attaching the other side.
DH bought one and put it up today, (he has had to take a little bit off the legs to get it flush with out bed) so tonight is the first night of trying it as a co-sleeper, I'll let you know how we get on if you are interested!